A modal logic amalgam of classical and intuitionistic propositional logic

We present a modal extension of classical propositional logic in which intuitionistic propositional logic is mirrored by means of the modal operator. In this sense, the modal extension combines classical and intuitionistic propositional logic avoiding the collapsing problem. Adopting ideas from a recent paper (S. Lewitzka, A denotational semantics for a Lewis-style modal system close to S1, 2013), we define a non-Fregean-style semantics such that propositional identity φ ≡ ψ is strict equivalence (φ→ ψ) ∧ (ψ → φ). It turns out that the modal operator , when restricted to propositional formulas, can be regarded as an intuitionistic truth predicate. As a main result, we prove that a propositional formula φ is an intuitionistic theorem iff φ is a theorem of our modal logic. Moreover, we show the existence of models which, for any propositional formula φ, satisfy φ iff φ is an intuitionistic theorem. In the context of such models, the modal operator can be seen as a predicate for intuitionistic validity. Finally, we characterize the class of models as the class of certain Heyting algebras with an operator.